Rigidity properties of CR embeddings into hyperquadrics

We study the rigidity of holomorphic mappings from a neighborhood of a Levi-nondegenerate CR hypersurface M with signature l into a hyperquadric Q_̂{l'}{N} in CP̂{N+1} of larger dimension and signature. Recent work of Baouendi, Ebenfelt, and Huang shows that if the difference in signature between a source manifold with positive CR complexity and target quadric is zero then a super- rigidity phenomenon holds. Another recent paper by the same authors shows that if the CR complexity is zero (and the difference in signature is nonnegative) then a partial rigidity phenomenon occurs. Our work considers both positive CR complexity and positive signature difference simultaneously, and we prove a partial rigidity result. Our main result is that if the CR complexity of M is not too large then the image of M under any such mapping is contained in a complex plane with a dimension depending only on the CR complexity and the signature difference, but not on N, the CR dimension of the target quadric. This result follows from two theorems, the first demonstrating that for sufficiently degenerate mappings, the image of M is contained in a plane, and the second relating the degeneracy of mappings into different quadric